Let's solve each problem by applying the properties of logarithms.

Properties to use:

1. $a \cdot \log(x) = \log(x^a)$
2. $\log(x) + \log(y) = \log(xy)$
3. $\log(x) - \log(y) = \log\left(\frac{x}{y}\right)$

Solution:

a. $5 \log(x) + 3 \log(z) - \frac{1}{2} \log(y)$

1. Apply the power rule:

   • $5 \log(x) = \log(x^5)$
   • $3 \log(z) = \log(z^3)$
   • $\frac{1}{2} \log(y) = \log(\sqrt{y})$

   So the expression becomes: $\log(x^5) + \log(z^3) - \log(\sqrt{y})$

2. Combine using the addition and subtraction properties: $\log\left(\frac{x^5 \cdot z^3}{\sqrt{y}}\right)$

Thus, the answer for (a) is: $\log\left(\frac{x^5 \cdot z^3}{\sqrt{y}}\right)$

b. $\frac{1}{2} \log(x) - 3 \log(y) - 5 \log(z)$

1. Apply the power rule:

   • $\frac{1}{2} \log(x) = \log(\sqrt{x})$
   • $3 \log(y) = \log(y^3)$
   • $5 \log(z) = \log(z^5)$

   So the expression becomes: $\log(\sqrt{x}) - \log(y^3) - \log(z^5)$

2. Combine using the subtraction property: $\log\left(\frac{\sqrt{x}}{y^3 \cdot z^5}\right)$

Thus, the answer for (b) is: $\log\left(\frac{\sqrt{x}}{y^3 \cdot z^5}\right)$

c. $\frac{3}{2} \log(x) - \frac{5}{2} \log(y) - \frac{3}{2} \log(z)$

1. Apply the power rule:

   • $\frac{3}{2} \log(x) = \log(x^{3/2}) = \log(\sqrt{x^3})$
   • $\frac{5}{2} \log(y) = \log(y^{5/2}) = \log(\sqrt{y^5})$
   • $\frac{3}{2} \log(z) = \log(z^{3/2}) = \log(\sqrt{z^3})$

   So the expression becomes: $\log(\sqrt{x^3}) - \log(\sqrt{y^5}) - \log(\sqrt{z^3})$

2. Combine using the subtraction property: $\log\left(\frac{\sqrt{x^3}}{\sqrt{y^5} \cdot \sqrt{z^3}}\right)$

Thus, the answer for (c) is: $\log\left(\frac{\sqrt{x^3}}{\sqrt{y^5} \cdot \sqrt{z^3}}\right)$

Let me know if you want further explanation on any step, or if you have additional questions!

Follow-Up Questions:

1. What are some common applications of logarithmic properties in mathematics?
2. How does the power rule for logarithms help simplify complex expressions?
3. Can logarithmic properties be applied in solving exponential equations? How?
4. What would be the result if the given expression had natural logarithms instead?
5. How do logarithmic and exponential functions relate to each other?

Tip:

Always check if an expression can be simplified further when using logarithmic properties to avoid errors in calculations.